We have reflected on several examples of reasoning and proof in the middle grades through problems in which students look for a pattern and then generalize that pattern into a rule for any number within that problem. In the Building Rafts with Rods problem, the students were able to generalize a formula to find the surface area for any number of rods. This is an example of algebraic reasoning. All aspects of mathematics involve reasoning and proof. In the middle grades, we want students to develop understanding of concepts through algebraic, geometric, proportional, probabilistic, and statistical reasoning.

The Reasoning and Proof Standard is not only part of each of the content standards, it also is closely related to the other process standards, as shown below.

Communication

A major part of the Reasoning and Proof Standard is clearly communicating one's ideas orally or in writing. This gives students opportunities to make their ideas understood; to justify, amend, and refine their thinking by presenting their own ideas; and to answer questions that require them to justify their thinking. Additionally, students have opportunities to solidify their thinking by listening to the ideas of others and asking carefully constructed questions.

Problem Solving

You will recall from the session on problem solving that as students work to solve rich mathematical problems, our expectation should be for them to include clear explanations as part of the problem-solving process. Those explanations must include reasons for why they did what they did, as well as support the reasonableness of their results. Problems should also be set in a context in which students have opportunities to generalize their ideas based on the relationships they find. This is accomplished by using a variety of strategies, such as making a table or solving a simpler problem. We saw a clear example of this in the Building Rafts with Rods problem.

Representation

Through the use of multiple representations, such as graphs, models, diagrams, and drawings, students clarify their thinking, develop sound reasoning, and often find ways to generalize what they have found to fit a variety of similar situations.

Connections

Sound mathematical reasoning and proof are often the result of students seeing connections among mathematical topics. Algebraic reasoning is frequently a result of visualizing a geometric situation. Proportional reasoning is the foundation for many geometric and statistical concepts.